Math problem set Grade 7 Engineering & Design
Ironmoss Creek Digester: Biogas Sizing and Solids Reduction
A grade 7 problem set on a fictional sludge digester where students work percent reduction, biogas volume, and the share of a plant's daily electricity it covers.
Objectives
- Students will calculate the mass destroyed and the mass remaining after a stated percent reduction.
- Students will convert a biogas volume into electricity and express it as a percent of a daily demand.
- Students will combine unit costs and rates across several steps to find a daily savings figure.
Vocabulary
Key words: digester, biogas, dry solids, percent reduction
Teacher overview
The Ironmoss Creek Water Resource Recovery Facility is a fictional wastewater plant used throughout this set. Its sludge is currently hauled away by truck, which is expensive, so the design team is studying an anaerobic digester. Bacteria inside a digester break down part of the solids without oxygen, which leaves less material to haul and produces a methane-rich gas the plant can burn for electricity. This set walks students through the solids balance, the gas volume, the electricity it yields, and the money the change would save. A common error in this math is treating "fifty-five percent destroyed" as "fifty-five percent left over," so students subtract in the wrong direction and report the destroyed mass as the mass still needing a truck.
Problems
Use these figures from the fictional Ironmoss Creek plant for every problem. The plant produces 4,000 pounds of dry solids each day. The proposed digester destroys 55 percent of those solids. Every pound of solids destroyed yields 12 cubic feet of biogas. Every 100 cubic feet of biogas run through the plant's engine makes 6 kilowatt-hours of electricity. The whole plant uses 9,000 kilowatt-hours each day.
How many pounds of dry solids does the digester destroy each day?
How many pounds of dry solids are left each day for the trucks to haul away?
How many cubic feet of biogas does the digester produce each day?
How many kilowatt-hours of electricity does that biogas make each day? Round to the nearest tenth.
What percent of the plant's daily electricity use does your answer to problem 4 cover? Round to the nearest tenth of a percent.
Hauling sludge costs the plant $0.06 for every pound. How much does the plant save each day on hauling, now that fewer pounds leave the site?
How much does the plant save on hauling across a 30-day month?
The plant buys electricity at $0.11 for each kilowatt-hour. What is the value of the electricity the biogas makes in one day? Round to the nearest cent.
What is the plant's total daily savings from hauling and electricity together?
The design team set this target for the project: the digester must cut the hauled solids by at least half. Does the design meet that target? Name one constraint the team must still check before building, and explain why it matters.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 2,200 pounds.
55 percent of 4,000 pounds = 0.55 x 4,000 = 2,200 pounds destroyed each day.Answer: 1,800 pounds.
4,000 pounds produced minus 2,200 pounds destroyed = 1,800 pounds left to haul. Students who answer 2,200 here have reversed the two figures.Answer: 26,400 cubic feet.
2,200 pounds destroyed x 12 cubic feet of biogas per pound = 26,400 cubic feet each day.Answer: 1,584 kilowatt-hours.
26,400 cubic feet divided by 100 = 264 groups of one hundred cubic feet. 264 x 6 kilowatt-hours = 1,584 kilowatt-hours.Answer: 17.6 percent.
1,584 divided by 9,000 = 0.176, which is 17.6 percent.Answer: 132 dollars.
The trucks now carry 2,200 fewer pounds each day. 2,200 pounds x $0.06 per pound = $132.00 saved per day.Answer: 3,960 dollars.
$132.00 per day x 30 days = $3,960.00 across the month.Answer: 174.24 dollars.
1,584 kilowatt-hours x $0.11 per kilowatt-hour = $174.24.Answer: 306.24 dollars.
$132.00 from hauling plus $174.24 from electricity = $306.24 in total daily savings.Answer: Yes, the design meets the target.
Problem 1 shows 2,200 of the 4,000 pounds destroyed, which is more than half, so the solids-reduction target is met. Accept any genuine constraint with a reason attached: the site must have room for a large sealed tank, methane is flammable and needs safe storage and burner controls, the digester must be kept warm and takes weeks to start up, the leftover liquid still has to be treated, or the purchase price has to fit the plant's capital budget.
Differentiation
Support
Give these students a filled-in solids diagram showing the 4,000 pounds entering, the destroyed share leaving as gas, and the remainder leaving by truck, so problems 1 and 2 become a labeling task rather than a memory task. Provide problem 4 as two separate boxes, one for the division by one hundred and one for the multiplication. Allow a calculator throughout.
Extension
Ask students to find how many pounds of dry solids the plant would have to produce each day for the biogas engine to cover one quarter of the plant's daily electricity use, working backward through the same chain of rates. Then have them explain in writing why a small plant is less likely than a large one to run a digester.
ELL supports
Pre-teach digester, solids, haul, and savings with a picture sequence: sludge tank, sealed digester, gas flame, truck. Provide the sentence frame "The digester destroys ___ pounds, so ___ pounds are left" for problems 1 and 2. Allow problem 10 to be answered with the frame "The design meets the target because ___, but the team must still check ___."
Extensions
Have students build a two-column table comparing the plant with and without the digester across four rows: pounds hauled each day, hauling cost each day, electricity bought each day, and net daily cost. Ask them to write one paragraph explaining which row changes most and why a plant manager might still hesitate. Students who finish early can estimate how many days of savings it would take to repay a purchase price the class agrees on together.